Adsorption kinetics from micellar solutions

So far, all theoretical models are based on surfactant solutions with a distribution of surfactant molecules as monomers. One of the specific properties of surfactants is that they form aggregates once a certain concentration, the critical micelle concentration CMC, is reached. The shape and size of such aggregates are different and depend on the structure and chain length of the molecules. At higher concentrations, far beyond the CMC, the phase behaviour is often complex giving rise to novel physical properties (Hoffmann 1990). 

The schematic shows that the transport of monomers and micelles as well as the mechanism of micelle kinetics have to be taken into account in a reasonable physical model. 

To account for the micelle effect, specific parameters of the respective surfactant micelles have to be known. Numerous papers on the determination of aggregation numbers and rate constants of micelle kinetics of many surfactants have been published (for example Aniansson et al. 1976, Hoffmann et al. 1976, Kahlweit Teubner 1980). Different micelle kinetics mechanisms exist, for example that summarised by Zana (1974). Three of these mechanisms are demonstrated in Fig. 4.12. 

The formation-dissolution mechanism assumes a total dissolution of a micelle in order to reestablish the local equilibrium monomer concentration. This model is based on an idealised distribution of only monomers and micelles with a definite aggregation number. Mechanism 2 is based on the existence of micelles of different size and therefore, a broad micelle size 

The physical model, based on the micelle kinetics mechanism 1, has the following form. The transport of monomers is given by. 

Surfactants form micelles above the critical micelle concentration (c.m.c.) of different sizes and shapes, depending on the nature of the molecule, temperature, electrolyte concentration, etc. (see Chapter 2). The dynamic nature of micellisation can be described by two main relaxation processes, ti (the life time of a monomer in a micelle) and t2 (the life time of the micelle, i.e. complete dissolution into monomers). 

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The first attempt to take into account the two-step kinetic theory of micellisation was made by Fainerman [147]. With that end in view two pairs of diffusion equations (for micelles and monomers) were written down for two situations eorresponding to the fast and slow proeesses. Approximate solutions of the boundary problems for these equations were used subsequently in the course of analysis of experimental data on the adsorption kinetics from micellar solutions [77, 85, 87, 88]. However, as it has been shown by Dushkin et al. [137], this approaeh is equivalent to the PFOR model for the slow proeess and probably eannot be applied to the description of the adsorption kinetics for the fast process. 

Another difficulty arising from this comparison is connected with the mathematical complexity of the corresponding boundary problems even if only linear diffusion equations are used. The mathematical description of the adsorption kinetics from micellar solutions is essentially more complicated in comparison with the case of the adsorption process from sub-micellar solutions. Analytical solutions of the corresponding boundary problems using rather poor approximations have been obtained only for a small number of situations. A sufficiently general solution cannot be obtained analytically and the deficiency of the rather well elaborated numerical methods often compel experimentalists to apply approximate solutions. Therefore, it seems important to consider the main equations proposed for the description of kinetic dependencies of the surface tension and adsorption, and to elucidate the limits of their application before the discussion of experimental results. 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

One of the reasons of the insufficient reliability of micellisation kinetics data determined from dynamic surface tensions, consists in the insufficient precision of the calculation methods for the adsorption kinetics from micellar solutions. It has been already noted that the assumption of a small deviation from equilibrium used at the derivation of Eq. (5.248) is not fulfilled by experiments. The assumptions of aggregation equilibrium or equal diffusion rates of micelles and monomers allow to obtain only rough estimates of the dynamic surface tension. An additional cause of these difficulties consists in the lack of reliable methods for surface tension measurements at small surface ages. The recent hydrodynamic analysis of the theoretical foundations of the oscillating jet and maximum bubble pressure methods has shown that using these techniques for measurements in the millisecond time scale requires to account for numerous hydrodynamic effects [105, 158, 159]. These effects are usually not taken into account by experimentalists, in particular in studies of micellar solutions. A detailed analysis of... 

This relation describes not only periodic deformations of a liquid surface. Using methods of integral transformations it is possible to show that the dynamic surface elasticity is a fundamental surface property and its value determines the system response to a small arbitrary surface dilation [161]. With this method it is also possible to determine the dynamic elasticity of liquid-liquid interfaces where the surfactant is soluble in both adjacent phases [133]. Moreover, similar transformations lead to an expression for the dynamic surface elasticity for the case when the mechanism of the slow step of micellisation is determined by scheme (5.185) or for frequencies corresponding to the fast step of micellisation [133,134]. However, as stated above, it is the slow process which mainly influences the adsorption kinetics from micellar solutions. 

Most of the traditional adsorption studies of surfactants correspond to dilute systems without aggregation in the bulk phase. At the same time micellar solutions are much more important from a practical point of view. To estimate the equilibrium adsorption, neglecting the effect of micelles can usually lead to reasonable results. However, the situation changes for nonequilibrium systems when the adsorption rate can increase by orders of magnitude when the of surfactant concentration is increased beyond the CMC. Current interest in the adsorption from micellar solutions is mainly caused by recent observations that the stability of foams and emulsions depends strongly on the concentration in the micellar region [1]. This effect can be explained by the influence of the micellisation rate on the adsorption kinetics. 

It is well-known that the adsorption kinetics from non-micellar solutions can be described mathematically by a corresponding boundary problem for the diffusion equation [104, 105]. In the case of micellar solutions the diffusion equation for monomers must contain terms taking into account the influence of micelles. The single diffusion equation of monomers must be replaced by a system of two equations (for monomers and micelles). At last, it is necessary to introduce an additional boundary condition, which takes into account that micelles are not surface active. This are all alterations in the formulation of the mathematical problem. However, it will be shown below that the new problem is essentially more complex and can be solved analytically only for very particular situations and after introduction of additional simplifications. 

Solutions of Triton X-100 were investigated most frequently probably because sufficient information on the equilibrium surface properties and adsorption kinetics from non-micellar... 

An analysis of the maximum bubble pressure method including all known theoretical approaches was given only recently so that data from literature are only of approximate character [160]. Therefore, the current level of kinetic theories of adsorption from micellar solutions and the corresponding experimental technique is still insufficient for investigations of the micellisation kinetics with a precision comparable to that of bulk relaxation methods. This pessimistic conclusion, however, relates to a less extent to methods based on small (mainly periodic) perturbations of the adsorption equilibrium. 

The equilibrium and dynamics of adsorption processes from micellar surfactant solutions are considered in Chapter 5. Different approaches (quasichemical and pseudophase) used to describe the micelle formation in equilibrium conditions are analysed. From this analysis relations are derived for the description of the micelle characteristics and equilibrium surface and interfacial tension of micellar solutions. Large attention is paid to the complicated problem, the micellation in surfactant mixtures. It is shown that in the transcritical concentration region the behaviour of surface tension can be quite diverse. The adsorption process in micellar systems is accompanied by the dissolution or formation of micelles. Therefore the kinetics of micelle formation and dissociation is analysed in detail. The considered models assume a fast process of monomer exchange and a slow variation of the micelle size. Examples of experimental dynamic surface tension and interface elasticity studies of micellar solutions are presented. It is shown that from these results one can conclude about the kinetics of dissociation of micelles. The problems and goals of capillary wave spectroscopy of micellar solutions are extensively discussed. This method is very efficient in the analysis of micellar systems, because the characteristic micellisation frequency is quite close to the frequency of capillary waves. 

Noskov, B.A., Kinetics of adsorption from micellar solutions, Adv. Colloid Interface Sci., 95, 237, 2002. 

In other experiments, Joos et al. [95,148] established that sometimes the dynamics of adsorption from micellar solutions exhibits a completely different kinetic pattern the interfacial relaxation is exponential, rather than inverse square root, as it should be for diffusion-limited kinetics. 

The Four Kinetic Regimes of Adsorption from Micellar Solutions In the theoretical model proposed in Refs. [149,150], the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers) is avoided. The theoretical problem is reduced to a system of four nonlinear differential equations. The model has been applied to the case of surfactant adsorption at a quiescent interface [150], that is, to the relaxation of surface tension and adsorption after a small initial perturbation. The perturbations in the basic parameters of the micellar solution are defined in the following way ... 

In summary, four distinct kinetic regimes of adsorption from micellar solutions exist, called AB, BC, CD, and DE see Figures 4.8 and 4.10. In regime AB, the fast micellar process governs the adsorption kinetics. In regime BC, the adsorption occurs under diffusion control because the... 

Bijsterbosch, H.D. Stuart, M.A.C. Fleer, G.J. Adsorption kinetics of diblock copolymers from a micellar solution on silica and titania. Macromolecules 1998, 31, 9281-9294. lijima, M. Nagasaki, Y. Okada, T. Kato, M. Kataoka, K. Core-polymerized reactive micellesm from heterotelechelic amphiphilic block copolymers. Macromolecules 1999, 32, 1140-1146. 

For small disturbances of the adsorption layer from equilibrium, Lucassen (1976) derived an analytical solution (cf Section 6.1.1). An analysis of the effect of a micellar kinetics mechanism of stepwise aggregation-disintegration and the role of polydispersity of micelles was made by Dushkin Ivanov (1991) and Dushkin et al. (1991). Although it results in analytical expressions, it is based on some restricting linearisations, for example with respect to adsorption isotherm, and therefore, it is valid only for states close to equilibrium. 

The region of the CMC (n (c +o c ) c l) requires special consideration. Substitution of of t2 and ti from Eq. (5.264) into (5.272), and the transition to the limit c -> 0 leads to the dynamic surface elasticity of sub-micellar solutions [165] and thus to a rather obvious conclusion if a solution contains mainly monomers, micelles do not influence the dynamic surface properties. Therefore, even for low frequencies (diffusion controlled adsorption kinetics) there is a concentration range close to the CMC where the surface elasticity is almost constant and begins to increase gradually only at further increasing concentration. Finally the surface elasticity takes values given by relations (5.275) - (5.278). This concentration dependence was observed in experiments with nonionic surfactants [95]. The oscillating barrier... 

When an adsorption layer is formed from a micellar solution, then monomers adsorb at the surface. This decreases the monomer concentration locally, the monomers and micelles are out of equilibrium and micelles will disintegrate. Hence, locally the concentration of micelles decreases and micelles will diffuse too. Thus, the presence of micelles in the solution bulk can be seen as extra source of matter, i.e. the micellar kinetics represents an additional relaxation mechanism to interfacial perturbations. 

As already mentioned, the surfactants are used to stabilize the liquid films in foams, in emulsions, on solid surfaces, and so forth. We will first consider the equilibrium and kinetic properties of surfactant adsorption monolayers. Various two-dimensional equations of state are discussed. The kinetics of surfactant adsorption is described in the cases of dijfusion and barrier control. Special attention is paid to the process of adsorption from ionic surfactant solutions. Theoretical models of the adsorption from micellar surfactant solutions are also presented. The rheological properties of the surfactant adsorption mono-layers, such as dilatational and shear surface viscosity and suiface elasticity, are introduced. The specificity of the proteins as high-molecular-weight surfactants is also discussed. 

As mentioned earlier, below we focus om attention on the kinetics of surfactant adsorption. First, we introduce the basic equations. Next, we consider the two alternative cases of surfactant adsorption under diffusion and barrier control. Special attention is paid to the adsorption of ionic surfactants, whose molecules are involved in long-range electrostatic interactions. Finally, we consider the adsorption from micellar surfactant solutions, which is accompanied by micelle diffusion, assembly, or disintegration. 

There are various direct measurements of micellar solutions giving access to the dynamics rate constants - mainly based on disturbance of the equilibrium state by imposing various types of perturbations, such as stop flow, ultrasound, temperature and pressure jump [14,15[. This aspect is also not further elaborated here we focus instead on the impact of micellar kinetics on interfacial properties, to demonstrate that tensiometry and dilational rheology are suitable methods to probe the impact of micellar dynamics. The first work on this subject was published by Lucassen already in 1975 [16[ and he showed that the presence of micelles in the bulk have a measurable impact on the adsorption kinetics, and hence on the dilational elasticity, when measured by a longitudinal wave damping technique. Subsequent work demonstrated the effect of micellar dynamics on non-equilibrium interfacial properties [17-29]. The physical idea of the impact of micellar dynamics on the dynamic properties of interfacial layers can be easily understood from the scheme given in Figure 13.1. 

Using the surfactant Ci4EOg we have demonstrated how the effect of micellar kinetics on the dynamic properties of adsorption layers can be experimentally studied. TUthough the simplest theoretical models are applied, a rather good agreement wdth the experimental data is observed. In adsorption dynamics from solutions far above the CMC we observe adsorption rates that cannot be explained by... 

In the acidic route (with pH < 2), both kinetic and thermodynamic controlling factors need to be considered. First, the acid catalysis speeds up the hydrolysis of silicon alkoxides. Second, the silica species in solution are positively charged as =SiOH2 (denoted as I+). Finally, the siloxane bond condensation rate is kinetically promoted near the micelle surface. The surfactant (S+)-silica interaction in S+X 11 is mediated by the counterion X-. The micelle-counterion interaction is in thermodynamic equilibrium. Thus the factors involved in determining the total rate of nanostructure formation are the counterion adsorption equilibrium of X on the micellar surface, surface-enhanced concentration of I+, and proton-catalysed silica condensation near the micellar surface. From consideration of the surfactant, the surfactants first form micelles as a combination of the S+X assemblies, which then form a liquid crystal with molecular silicate species, and finally the mesoporous material is formed through inorganic polymerization and condensation of the silicate species. In the S+X I+ model, the surfactant-to-counteranion... 

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